Journal Article

Mathematical modelling of avascular-tumour growth

J. P. WARD and J. R. KING

in Mathematical Medicine and Biology: A Journal of the IMA

Published on behalf of Institute of Mathematics and its Applications

Volume 14, issue 1, pages 39-69
Published in print March 1997 | ISSN: 1477-8599
Published online March 1997 | e-ISSN: 1477-8602 | DOI: http://dx.doi.org/10.1093/imammb/14.1.39
Mathematical modelling of avascular-tumour growth

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A system of nonlinear partial differential equations is proposed as a model for the growth of an avascular-tumour spheroid. The model assumes a continuum of cells in two states, living or dead, and, depending on the concentration of a generic nutrient, the live cells may reproduce (expanding the tumour) or die (causing contraction). These volume changes resulting from cell birth and death generate a velocity field within the spheroid. Numerical solutions of the model reveal that after a period of time the variables settle to a constant profile propagating at a fixed speed. The travelling-wave limit is formulated and analytical solutions are found for a particular case. Numerical results for more general parameters compare well with these analytical solutions. Asymptotic techniques are applied to the physically relevant case of a small death rate, revealing two phases of growth retardation from the initial exponential growth, the first of which is due to nutrient-diffusion limitations and the second to contraction during necrosis. In this limit, maximal and ‘linear’ phase growth speeds can be evaluated in terms of the model parameters.

Keywords: tumour growth; avascular; mathematical modelling; numerical solution; asymptotic analysis

Journal Article.  0 words. 

Subjects: Applied Mathematics ; Biomathematics and Statistics

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